category: literaturenote citekey: chiriconvexoptimizationboundarycontrol2025 title: Convex Optimization Boundary-Control Policies for Stability and Invariance of Traffic Flow Dynamics authors: "Chiri, Maria Teresa; Guglielmi, Roberto; Notomista, Gennaro" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2503.14666v1" zotero_key: GKGIZ236 zotero_storage: I4WSCREV collections: transpordi_planeerimine folder: Liiklussageduse kaudne hindamine/05_Artiklid firstAuthor: "Chiri, Maria Teresa"

status: converted

Convex Optimization Boundary-Control Policies for Stability and Invariance of Traffic Flow Dynamics

Maria Teresa Chiri, Roberto Guglielmi, and Gennaro Notomista

*Abstract*— In this letter we propose an optimization-based boundary controller for traffic flow dynamics capable of achieving both stability and invariance conditions. The approach is based on the definition of Boundary Control Barrier Functionals (BCBFal), from which sets of invariance-preserving boundary controllers are derived. In combination with sets of stabilizing controllers, we reformulate the problem as a convex optimization program solved at each point in time to synthesize the boundary control inputs. Sufficient conditions for the existence of optimal controllers are derived in the case of traffic flow models with stability and invariance conditions.

I. INTRODUCTION

Applications of control of Partial Differential Equations (PDEs) arise in several different fields, including fluid, thermal, structural, and delay systems. Typical control objectives in such scenarios include stability and invariance. The former encodes long-term objectives such as eventually steering the system to a desired state (e.g. achieving a desired temperature profile of a metal plate), while the latter is concerned with preventing the system from exiting a desired safe set (e.g. limiting the maximum stress in an Euler-Bernoulli beam below the yield point).

Considerable attention has been given to the synthesis of stabilizing controllers for PDEs, which has led to the development of approaches such as control Lyapunov functionals and backstepping. The latter has been shown to be useful in traffic, thermal, and multi-agent control, among other applications. Yet, the practical implementation of such controllers requires resorting to numerical approximation and ad-hoc computational techniques [1]. Invariance has been considered in the context of optimization-based controllers and is typically encoded as state constraints to be satisfied over the optimization horizon [2], [3]. Necessary conditions are derived for the existence and regularity of the controller. Nevertheless, the computational complexity of existing algorithms is such that they cannot be implemented in online settings, especially in most applications of interest, including fluid dynamics and traffic control.

In this letter, we propose an optimization-based boundary controller for traffic flow dynamics capable of holistically ensuring stability and invariance conditions. The approach is

Maria Teresa Chiri is with the department of Mathematics and Statistics, Queen's University, Kingston, ON, Canada. maria.chiri@queensu.ca

Roberto Guglielmi is with the department of Applied Mathematics, University of Waterloo, ON, Canada roberto.guglielmi@uwaterloo.ca

Gennaro Notomista is with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada. gennaro.notomista@uwaterloo.ca

inspired by the recent developments in stabilizing and safetypreserving controllers in the ordinary differential equations literature, where control Lyapunov [4] and control barrier functions [5] have been unified in a quadratic programming formulation, in order to achieve stability and invariance, respectively. Starting from the definition of Boundary Control Barrier Functionals (BCBFal) for a traffic flow model, we derive constraints on the boundary controllers that ensure the verification of invariance-like conditions, expressed as the positivity of the zero superlevel set of continuously differentiable functionals. These constraints are then enforced in a convex optimization program whose solution—computable in polynomial time—is a stabilizing and invariance-preserving boundary controller.

Approaches similar in nature to what we propose in this letter have been recently developed in works such as [6] and [7]. These works consider the use of barrier functions in the context of PDEs and, while the former leverages neural operators for PDE control [8], the latter focuses on hyperbolic PDE-ODE cascades. Differently from these approaches, our proposed optimization-based control synthesis framework does not need to learn a representation of inputoutput mappings and is able to enforce constraints directly on the state, rather than on the output, of a PDE model. As a matter of fact, we utilize the whole PDE dynamics in order to define the sets of boundary controllers that lead to the satisfaction of invariance constraints. Moreover, we formulate a computationally-efficient convex optimization problem—and analyze its feasibility—in order to synthesize the boundary control inputs to satisfy stability and invariance conditions simultaneously.

To summarize, the contributions of our work are the following: (i) We introduce the notion of boundary control barrier functionals for PDEs; (ii) We derive constraints on boundary control inputs in order for the state to satisfy invariance conditions; (iii) We formulate a convex optimization control policy to synthesize boundary control input to achieve stability and invariance conditions simultaneously; (iv) We provide sufficient conditions for the existence of such optimal boundary controllers.

II. TRAFFIC MODELING AND STABILIZING CONTROL

A. Boundary value problem for the LWR Model

The Lighthill-Whitham-Richards (LWR) model [9], [10] is a fundamental macroscopic model for traffic flow which describes the conservation of the total number of vehicles and assumes that the average traffic speed V (x, t) depends solely on the traffic density u(x, t). Consequently, the mean traffic flow (i.e., the number of vehicles crossing point x per unit time) is given by f(x, t) = u(x, t)V (u(x, t)), leading to the hyperbolic conservation law u^t + (uV (u))^x = 0.

In particular, the velocity function attains its maximum when the vehicles density is zero and decreases to zero as the density increases. Specifically, we will assume that

$$f: [0, u{\text{max}}] \to \mathbb{R}+, \quad f(u) = u \left(1 - \frac{u}{u_{\text{max}}}\right)$$

and denote with uˆ = umax/2 the unique point of maximum. The initial-boundary value problem (IBVP) for the LWR model on the interval (a, b) with initial data u⁰ is given by

$$u_t + f(u)_x = 0,$$ $u(0, x) = u_0(x),$ (1)

$$u(t,a) = \omega_a(x), \qquad u(t,b) = \omega_b(x).$$ (2)

In the remainder of the letter, we will consider weak entropy solutions for the IBVP with BV data, i.e., weak solutions on the interval [a, b] that satisfy Lax admissibility conditions and the weak boundary conditions introduced in [11].

B. Lyapunov-based Stabilization

The Lyapunov boundary stabilization problem for the IBVP (1)-(2) can be formulated as follows:

Problem 1. Given a stationary solution u ∗ to the conservation law (1), do there exist boundary conditions ω^a and ω^b such that the IBVP is well posed and its solution is stable in the sense of Lyapunov at u ∗?

To address this problem, we can follow [12] where the authors consider the Lyapunov function

$$V(u(t)) = \frac{1}{2} \int_{a}^{b} (u(t, x) - u^{*})^{2} dx$$ (3)

with u weak entropy solution to (1). Continuity of the mapping t → u(t, ·) from [0, T] to L 1 (a, b) with T > 0 ensures that the function V (·) is well-defined and continuous. We can assume that u is piecewise smooth as solutions to (1)-(2) with initial condition in BV can be approximated by the solution to an IBVP with piecewise smooth solution at all times (see [12]). We index the jump discontinuities of u(t) in increasing order of their location at time t by i = 0, . . . , N(t), including the boundaries a, b with x0(t) = a and x^N (t) = b and write:

$$V(u(t)) = \frac{1}{2} \sum{i=0}^{N(t)-1} \int{xi(t)}^{x{i+1}(t)} (u(t,x) - u^*)^2$$

Then differentiating we get

$$\frac{dV}{dt}(t) = (u(t,a) - u^)f(u(t,a)) - (u(t,b) - u^)f(u(t,b))

F(u(t,a)) + F(u(t,b))
\sum_{i=1}^{N(t)-1} \Delta_i((u-u^*)f(u) - F(u))
\sum_{i=1}^{N(t)-1} \frac{(u(t,x_i^-) - u^) + (u(t,x_i^+) - u^)}{2}$$ (4)

where F is a primitive of the flux f. The last two terms of (4) correspond to jump discontinuities in the solution. While they are neither observable nor controllable from the boundaries, they contribute to the decrease of the Lyapunov function [12, Proposition 1]. Therefore, stabilizing boundary controls should be designed to ensure that

$$(u(t,a) - u^) f(u(t,a)) - (u(t,b) - u^) f(u(t,b)) - F(u(t,a)) + F(u(t,b)) \le 0.$$

Previous control strategies proposed in the literature are greedy, non-local, and formulated based on the boundary traces of the solutions and the initial data. In the next section, however, we propose boundary controls derived as solutions to optimization problems that consider the whole dynamics of the conservation law.

III. OPTIMIZATION-BASED TRAFFIC CONTROL VIA CONTROL LYAPUNOV AND BARRIER FUNCTIONALS

The motivation of the optimization-based boundary controller proposed in this letter originates from the following Problem, which, together with Problem 1, delineate two control objectives typically arising in the control of conservation laws in traffic applications.

Problem 2. Given a maximum value, u¯, of the solution to the conservation law (1), do there exist boundary conditions ω^a and ω^b such that the IBVP is well-posed and its solution u remains smaller than u¯ (measured in a suitable norm)?

Our approach to solve this problem, in combination with Problem 1, is based on the concept of boundary control barrier functionals for our problem, which will serve as the fundamental building block for determining optimal controls.

Definition 1 (Boundary Control Barrier Functional (BCB-Fal)). Let S ⊂ L (a, b) be the zero superlevel set of the functional b : S → R. b is a Boundary Control Barrier Functional (BCBFal) for the system (1)–(2) if there exist a class K function α such that

$$\sup_{\omega_a,\omegab \in [0,u{\max}]} \left{ -\frac{\partial b}{\partial u} f_x(u(t),\omega_a,\omega_b) + \alpha(b(u(t)) \right} \geq 0$$

for all u ∈ S.

Theorem 1. Let Ω^u := {(s, z) : − ∂u fx(u(t), s, z) + α(b(u(t)) ≥ 0} where b is a BCBFal. Any BV boundary controllers (ωa, ωb) ∈ Ω^u renders the set S forward invariant, i.e. u(t) ∈ S for all t ≥ 0.

Proof. If follows from the application of the comparison lemma to the ordinary differential equation ˙b+α(b) = 0.

Remark 1. Despite the similarity in the name, the BCBFal introduced in Definition 1 differs from the neural BCBF introduced in [6] in the following fundamental aspects. First, the control objective in [6] is defined for the trace of the state u(0, t), rather than the state function u itself. That also explains why we call it a functional. Second, while the authors in [6] leverage neural network representations to overcome the non-Markov nature of the problem they consider, acting on the whole state via BCBFals we are able to enforce conditions on the boundary control that lead to state constraint verification without the need of introducing learned components.

In the remainder, we consider the following BCBFal:

$$b(u(t)) = \bar{u}^2 - \int_a^b u(t, x)^2 dx,$$ (5)

whose zero superlevel set corresponds to the states u with L -norm less than a threshold u¯, as prescribed in the control objectives described in Problem 2.

A. Convex Optimization Control Policies

We are interested in convexity for the sake of computational efficiency. Specifically, we aim to identify intervals where V˙ is convex with respect to u(t, a) and u(t, b) as well as intervals where −˙b is convex in u(t, a) and u(t, b). It is straightforward to verify that:

V˙ convex in u(t, a) on C^a := [0, 2u ^∗+umax 4
V˙ convex in u(t, b) on C^b := [ ²^u ^∗+umax 4 , umax]
−˙b convex in u(t, a) on I^a := [0, umax 4
−˙b convex in u(t, b) on I^b := [ ^umax 4 , umax]
V˙ and −˙b convex in u(t, a) on [0, umax 4
V˙ and −˙b convex in u(t, b) on [ ^∗+umax , umax]

For other expressions of V and h we can optimize over different controllers to achieve different objectives, e.g.

$$\begin{cases} u(t,a) \in [0,\frac{2u^*+u{\max}}{4}] & \text{for stability} \ u(t,b) \in [\frac{u{\max}}{4},u_{\max}] & \text{for invariance} \end{cases}$$ (6)

$$\begin{cases} u(t,a) \in [0,\frac{u{\max}}{4}] & \text{for invariance} \ u(t,b) \in [\frac{2u^* + u{\max}}{4}, u_{\max}] & \text{for stability} \end{cases}$$ (7)

or even decide which boundary control to use online based on other metrics such as feasibility and norm of the control.

In the sequel, we introduce the notation

$$\begin{split} g(s,z) &:= (s-u^)f(s) - (z-u^)f(z) - F(s) + F(z), \ k(s,z) &:= sf(s) - zf(z) - F(s) + F(z), \ C(t) &= \alpha \left( V(u(t)) \right) \geq 0, \quad \alpha \in \mathcal{K}, \ D(t) &= \beta \left( b(u(t)) \right) \geq 0, \quad \beta \in \mathcal{K}. \end{split}$$

The formulation for the controller synthesis to address Problem 1 is achieved via one of the following convex optimization problems:

$$\min_{\omega_a(t) \in \mathcal{C}_a} \left{ \omega_a(t)^2 : g(\omega_a(t), u(t, b)) \le -C(t) \right}$$ (8)

$$\min_{\omega_b(t)\in\mathcal{C}_b} \left{ \omega_b(t)^2 : g(u(t,a),\omega_b(t)) \le -C(t) \right}$$ (9)

$$\min_{\substack{\omega_a(t) \in c_a \ \omega_b(t) \in c_b}} \left{ \omega_a(t)^2 + \omega_b(t)^2 : g(\omega_a(t), \omega_b(t)) \le -C(t) \right}$$ (10)

where stability is sought after via the left boundary, the right one, or both, respectively.

Similarly, to tackle Problem 2, the boundary controls are evaluated as the solutions of one of the following convex optimizations:

$$\min_{\omega_a(t)\in\mathcal{I}_a} \left{ \omega_a(t)^2 : k(\omega_a(t), u(t, b)) \le D(t) \right}$$ (11)

$$\min_{\omega_b(t)\in\mathcal{I}_b} \left{ \omega_b(t)^2 : k(u(t,a),\omega_b(t)) \le D(t) \right}$$ (12)

$$\min_{\substack{\omega_a(t) \in \mathcal{I}_a \ \omega_b(t) \in \mathcal{I}_b}} \left{ \omega_a(t)^2 + \omega_b(t)^2 : k(\omega_a(t), \omega_b(t)) \le D(t) \right}$$ (13)

that seek for the optimal control that ensures invariance via the left boundary, the right one, or both, respectively.

Theorem 2. There exist values of the parameters u ∗ , u¯ ≤ umax and functions α, β ∈ K such that the optimization problems (8), (9), (10), (11), (12), (13) are feasible.

The proof is constructive and it is illustrated in the following section. It is worth noticing that we can deal with compound problems too, where one boundary control aims to stabilize the system and the other boundary control ensures that the invariance condition is preserved, as in (6) and (7). This will be addressed in Theorem 3 in Section V.

IV. EXISTENCE OF OPTIMAL CONTROLLERS

A. Stabilizing Controllers

Let us start with problem (8):

$$\min_{\omega_a(t) \in \mathcal{C}_a} \left{ \omega_a(t)^2 : g(\omega_a(t), u(t, b)) \le -C(t) \right}.$$

This is a constrained minimization problem; therefore, the optimal control is either the trivial one, ωa(t) = 0, provided that it belongs to the region identified by the constraint, or it lies on the boundary of the region defined by the constraint, i. e. ωa(t) is a root of the nonlinear equation

$$g(\omega_a(t), u(t, b)) = -C(t). \tag{14}$$

Let p(s) := g(s, u(t, b)), δ = min{u ∗ , uˆ}, γ = max{u ∗ , uˆ}. If u ^∗ ̸= ˆu, then p ′ (s) < 0 on [0, δ)∪(γ, umax] and p ′ (s) > 0 on (δ, γ). Since ²^u ^∗+umax ⁴ = u ^∗+ˆu , equation (14) has at most two roots on Ca. In particular, we have three different cases: if p(0) ≤ −C(t) the optimal control is given by ω ∗ a (t) = 0; if p(δ) ≤ −C(t) < p(0), the optimal control is given by the minimal root of equation (14); while if −C(t) < p(δ), equation (14) has no solution on Ca, and so (8) admits no solution, as illustrated in the left figure in Fig. 1.

If u ^∗ = ˆu, then p ′ (s) < 0 on [0, umax] \ {uˆ}, p ′ (ˆu) = 0. Thus, equation (14) has at most one root. In particular, if p(0) ≤ −C(t) the optimal control is given by ω ∗ a (t) = 0, if p( ^∗+umax ) ≤ −C(t) < p(0), the optimal control is given by the unique root of equation (14) on 0, 2u ^∗+umax 4 , while if −C(t) < p( 2u ^∗+umax 4 ), equation (14) has no solution on Ca, and so (8) admits no solution.

We now consider (9)

$$\min_{\omega_b(t) \in \mathcal{C}_b} \left{ \omega_b(t)^2 : g(u(t, a), \omega_b(t)) \le -C(t) \right}.$$

Fig. 1. On the left, a graphical representation of the roots of (14) associated with (8), case u ^∗ ̸= ˆu, for three different values of −C(t); on the right, a possible configuration of the curves for problem (10), first (blue) and second (green) curves from (15).

Let q(z) := g(s, z), and notice that q ′ (z) = ¹ (z−u ∗ )(z−uˆ). If u ^∗ ̸= ˆu, then then q ′ (z) > 0 on [0, δ) ∪ (γ, umax] and q ′ (z) < 0 on (δ, γ). Therefore, if −C(t) ≥ q 2u ^∗+umax 4 , ω ∗ b (t) = ²^u ^∗+umax 4 is the optimal solution to (9); if q(γ) ≤ −C(t) < q 2u ^∗+umax 4 , then ω ∗ b (t) is the minimal root of q(z) = −C(t) on 2u ^∗+umax 4 , umaxⁱ ; finally, if −C(t) < q(γ), then there exists no solution to (9). When u ^∗ = ˆu, then q ′ (z) = ¹ uˆ (z − uˆ) ² ≥ 0, and thus 2u

if −C(t) ≥ q ^∗+umax 4 , then ω ∗ b (t) = ²^u ^∗+umax 4 is the optimal solution to (9); if −C(t) < q 2u ^∗+umax 4 , then there exists no solution to (9).

We now look at (10):

$$\min_{\substack{\omega_a(t) \in \mathcal{C}_a \ \omega_b(t) \in \mathcal{C}_b}} \left{ \omega_a(t)^2 + \omega_b(t)^2 : g(\omega_a(t), \omega_b(t)) \le -C(t) \right}.$$

By taking the Lagrangian, we can observe that critical points are solutions of the system

$$\begin{cases} sz(f'(s) + f'(z)) = u^*(sf'(z) + zf'(s)) \ g(\omega_a(t), \omega_b(t)) + C(t) = 0. \end{cases}$$ (15)

An admissible optimal control, if any, is the point of minimal norm in the intersection of the curves defined by the equations in (15) (see right figure in Fig. 1). Due to the strong nonlinearity of these equations, we rely on numerical evidence to assert that an optimal control exists only if

$$C(t) = \alpha \left(\frac{1}{2} \int_a^b (u(t, x) - u^*)^2 dx\right) \approx 0$$

meaning that the solution of the boundary value problem is nearly at the target state.

Remark 2. Controlling from both ends appears significantly less effective—both in theory and simulations—than controlling from a single boundary point. For a comparison, see also the non-local two-boundary control in [12].

B. Invariance-preserving Controllers

Let us first consider the existence of solutions to the invariance problem (11), which reads as

$$\min_{\omega_a(t) \in \mathcal{I}_a} \left{ \omega_a(t)^2 : k(\omega_a(t), u(t, b)) \le D(t) \right}$$

The optimal solution is either ω ∗ a (t) = 0, provided that it satisfies the constraint, or it lies on the boundary of the region defined by the constraint, i.e.,

$$k(\omega_a^*(t),u(t,b))=D(t).$$

Let ℓ(s) := k(s, u(t, b)). It is easy to check that ℓ ′ (s) > 0 on Ia. So, if ℓ(0) ≤ D(t), then ω ∗ a (t) = 0 is the optimal solution to (11) and if D(t) < ℓ(0), then there exists no solution to (11).

Let us now consider problem (12):

$$\min_{\omega_b(t)\in\mathcal{I}_b} \left{ \omega_b(t)^2 : k(u(t,a),\omega_b(t)) \le D(t) \right}.$$

In this case, the optimal solution is ω ∗ b (t) = ^umax 4 , provided that it satisfies the constraint, or it lies on the boundary of the region defined by the constraint

$$k(u(t,a),\omega_b^*(t)) = D(t). \tag{16}$$

Let ρ(z) := k(u(t, a), z). We can then check that ρ (z) < 0 on [0, uˆ) and ρ ′ (z) > 0 on (ˆu, umax]. Therefore, (16) has at most two roots on Ib, and we conclude that: if D(t) ≥ ρ umax 4 , then ω ∗ b (t) = ^umax 4 is the optimal solution to (12); if ρ (ˆu) ≤ D(t) < ρ umax , then the optimal solution ω ∗ b (t) to (12) is given by the solution to (16) of minimal norm on Ib, and if D(t) < ρ (ˆu), then there exists no solution to (12).

We skip here the details of problem (13), which can be treated similarly to (10).

V. EXISTENCE OF STABILIZING AND INVARIANCE-PRESERVING CONTROLLERS

We now consider the compound problem that aims to ensure stability from one boundary point and invariance from the other one. With the insights gained in the previous section when studying the feasibility of (10) and (13), we reformulate (8)-(12) in the following relaxed version:

$$\min_{\omega_a(t) \in \mathcal{C}_a} \left{ \omega_a(t)^2 : g(\omega_a(t), z_1) \le -C(t), \ k(\omega_a(t), z_2) \le D(t), z_1, z_2 \in \mathcal{I}_b \right}$$ (17)

and

$$\min_{\omega_b(t)\in\mathcal{I}_b} \left{ \omega_b(t)^2 : g(s_1, \omega_b(t)) \le -C(t), \ k(s_2, \omega_b(t)) \le D(t), s_1, s_2 \in \mathcal{C}_a \right}.$$ (18)

Notice that we solve independently the two optimization problems yet ensuring that each satisfies both constraints within the admissibility class of the other.

Theorem 3. There exist values of the parameters u ∗ , u¯ ≤ umax and functions α, β ∈ K such that the optimization problem (17)-(18) is feasible.

Proof. We first focus on (17). It is clear that ω ∗ a (t) = 0 is the optimal solution provided that the constraints are satisfied. If not, then ω ∗ a (t) lies at the boundary of the constraint set, which is given by either

$$\begin{cases} g(\omega_a^(t), z_1) = -C(t), & z_1 \in \mathcal{I}_b, \ k(\omega_a^(t), z_2) \le D(t), & z_2 \in \mathcal{I}_b, \end{cases}$$ (19)

$$\begin{cases} g(\omega_a^(t), z_1) \le -C(t), & z_1 \in \mathcal{I}_b, \ k(\omega_a^(t), z_2) = D(t), & z_2 \in \mathcal{I}_b. \end{cases}$$ (20)

Denote by V the set of all admissible solutions to both (19) and (20) on Ca. We now discuss each case separately. Reasoning as in IV-A with the monotonicity of p(s) := g(s, z), z ∈ Ib, we characterize the solutions to (19) as follows.

Case u ^∗ ̸= ˆu: then p ′ (s) < 0 on [0, δ) ∪ (γ, umax] and p ′ (s) > 0 on (δ, γ). Thus,

a) If g(0, z1) ≤ −C(t) for some z¹ ∈ Ib, then ω ∗ a (t) = 0 is an admissible solution of (19) provided that k(ω ∗ a (t), z2) ≤ D(t) for some z² ∈ Ib. If so, we store it in V
b) If g(δ, z1) ≤ −C(t) < g(0, z1) for some z¹ ∈ Ib, then among all the roots s ∗ to p(s) = −C(t) select the one of minimal norm that also satisfies k(s ∗ , z) ≤ D(t) for some z ∈ Ib, if any, and store it in V
c) If −C(t) < g(δ, z) for all z ∈ Ib, then (19) has no solution and we proceed searching for a solution to (20)

Case u ^∗ = ˆu: p ′ (s) < 0 on [0, umax] \ {uˆ}, p ′ (ˆu) = 0. Thus,

a) If g(0, z1) ≤ −C(t) for some z¹ ∈ Ib, then ω ∗ a (t) = 0 is an admissible solution to (19) provided that k(ω ∗ a (t), z2) ≤ D(t) for some z² ∈ Ib, and we store it in V
b) If g( 2u ^∗+umax 4 , z1) ≤ −C(t) < g(0, z1) for some z¹ ∈ Ib, an admissible solution to (19) is given by the unique root s ^∗ of p(s) = −C(t) on 0, ^∗+umax 4 provided that k(s ∗ , z2) ≤ D(t) for some z² ∈ Ib, and we store it in V
c) If −C(t) < g( 2u ^∗+umax 4 , z) for all z ∈ Ib, equation p(s) = −C(t) has no root on Ca, and so (19) admits no solution and we proceed searching for a solution to (20)

We now consider the case (20). Arguing as in IV-B with the monotonicity of ℓ(s) := k(s, z), z ∈ Ib, we notice that ℓ ′ (s) > 0 on [0, umax/2) and ℓ ′ (s) < 0 on (umax/2, umax]. Since ²^u ^∗+umax ⁴ < umax , we get that

a) If D(t) ≥ k(0, z1) for some z¹ ∈ Ib, then ω ∗ a (t) = 0 is an admissible solution of (20) provided that g(ω ∗ a (t), z2) ≤ −C(t) for some z² ∈ Ib, and we store it in V
b) If D(t) < k(0, z1) for all z¹ ∈ Ib, (20) has no solution At this stage, the optimal solution to problem (17) is given by the element s ∗ in V of minimal norm.

In a similar manner, we can solve the problem (18). It is clear that ω ∗ b (t) = ^umax 4 is the optimal solution provided that the constraints are satisfied. If not, then ω ∗ b (t) lies at the boundary of the constraint set, which is given by either

$$\begin{cases} g(s_1, \omega_b^(t)) = -C(t), & s_1 \in \mathcal{C}_a \ k(s_2, \omega_b^(t)) \le D(t), & s_2 \in \mathcal{C}_a \end{cases}$$ (21)

$$\begin{cases} g(s_1, \omega_b^(t)) \le -C(t), & s_1 \in \mathcal{C}_a \ k(s_2, \omega_b^(t)) = D(t), & s_2 \in \mathcal{C}_a. \end{cases}$$ (22)

Denote by W the set of all admissible solutions to both (21) and (22) on Ib. In the case (21), arguing as in Section IV-A with the monotonicity of q(z) := g(s, z), s ∈ Ca, we have the following.

Case u ^∗ ̸= ˆu: then q ′ (z) > 0 on [0, δ) ∪ (γ, umax] and q ′ (z) < 0 on (δ, γ). Thus, if u ^∗ ≤ umax/4,

a) If g(s1, umax/4) ≤ −C(t) for some s¹ ∈ Ca, then ω ∗ b (t) = umax/4 is an admissible solution to (21) provided that k(s2, ω^∗ b (t)) ≤ D(t) for some s² ∈ Ca. If so, we store it in W
b) If g(s1, uˆ) ≤ −C(t) < g(s1, umax/4) for some s¹ ∈ Ca, then among all the roots z ∗ to q(z) = −C(t) select the one of minimal norm that also satisfies k(s, z^∗ ) ≤ D(t) for some s ∈ Ca, if any, and store it in W
c) If −C(t) < g(s, uˆ) for all s ∈ Ca, then (21) does not admit any solution on I^b and we proceed searching for a solution to the case (22)

If umax/4 < u^∗ < uˆ or u < u ˆ ∗ , then

a) If g(s1, umax/4) ≤ −C(t) for some s¹ ∈ Ca, ω ∗ b (t) = umax/4 is an admissible solution of (21) provided that k(s2, ω^∗ b (t)) ≤ D(t) for some s² ∈ Ca. If so, we store it in W
b) If g(s1, γ) < g(s1, umax/4) and g(s1, γ) ≤ −C(t) < g(s1, umax/4) for some s¹ ∈ Ca, then among the two roots z ∗ to q(z) = −C(t) select the one of minimal norm that also satisfies k(s2, z^∗ ) ≤ D(t) for some s² ∈ Ca, if any, and store it in W
c) If −C(t) < min{g(s, γ), g(s, umax/4)} for all s ∈ Ca, then (21) does not admit any solution on I^b and we proceed searching for a solution to the case (22)

Case u ^∗ = ˆu: since q ′ (z) = ¹ uˆ (z − uˆ) ² ≥ 0, thus

a) If g(s1, umax/4) ≤ −C(t) for some s¹ ∈ Ca, then ω ∗ b (t) = umax/4 is an admissible solution to (21) provided that k(s2, ω^∗ b (t)) ≤ D(t) for some s² ∈ Ca. If so, we store it in W
b) If −C(t) < g(s, umax/4) for all s ∈ Ca, then (21) does not admit any solution on I^b and we proceed searching for a solution to the case (22)

We now focus on the case (22). Reasoning as in Section IV-B with the monotonicity of ρ(z) := k(s, z), s ∈ Ca, we have the following.

a) If k(s2, umax/4) ≤ D(t) for some s² ∈ Ca, then ω ∗ b (t) = umax/4 is an admissible solution of (22) provided that g(s1, ω^∗ b (t)) ≤ D(t) for some s¹ ∈ Ca. If so, we store it in W
b) If k(s2, uˆ) ≤ D(t) < k(s2, umax/4) for some s² ∈ Ca, then among the two roots z ∗ to ρ(z) = D(t) select the one of minimal norm that also satisfies g(s1, z^∗ ) ≤ −C(t) for some s¹ ∈ Ca, if any, and store it in W
c) If D(t) < k(s, uˆ) for all s ∈ Ca, then (22) does not admit any solution on I^b

At this point, the optimal solution to problem (18) is given by the element z ∗ in W of minimal norm.

Finally, the solution to the compound problem is given by the pair (ω ∗ a (t), ω^∗ b (t)) where ω ∗ a (t) solves (17) and ω ∗ b (t) is the solution to (18), if both V and W are nonempty.

Fig. 2. Simulation results: the dashed lines correspond to the case of stability-only control (8), while the solid lines are for the stability and invariancepreserving control (8)-(12).

VI. SIMULATIONS

In this section, we showcase the behavior of system (1)- (2) with the proposed boundary controllers computed using (17) and (18) in order to achieve stability and invariance conditions, respectively. The parameters used to simulate the system are as follows: a = 0, b = 1, umax = 1, u ^∗ = 0.5 in (3), and u¯ = 0.4 in (5). We simulate the system over the time interval [0, 30]. To better illustrate the effect of the controller, we compare the stabilizing and invariance-preserving controller with a Lyapunov-based stabilizing boundary controller. The latter is computed as the solution of (8). This way, the first simulation controls only the left boundary to achieve stability, while the second controls also the right boundary in order to achieve invariance.

The snapshots of the two simulations are shown in Fig. 2. Figures 2a–2f report the time evolution of the state of the system (1). The red dashed line depicts the case where only ω^a is controlled, using (8), while the blue solid line represents the system state behavior under the stability and invariance-preserving boundary controllers achieved via ωaand ω^b computed by (17) and (18), respectively. In the first case, the stabilizing controller leads to state to settle at u ^∗ = 0.5. While in the second case, the invariance constraint that the L -norm of the state has to be less than u¯ = 0.4 is achieved. The invariance is obtained at the expense of stability, as can be quantitatively seen in Fig. 2g. Lower values of the Lyapunov function (green) correspond to the state u closer to the desired value u ∗ in the L 2 -norm sense, and positive values of the BCBFal mean that the norm of the state is less than u¯ (see, in particular, the zoomed in popout axes in Fig. 2g). The regularity of the controllers to achieve the desired behaviors can be appreciated in Fig. 2h, corroborating the choice made in Section V of solving (17) and (18) in place of (8)-(12).

VII. CONCLUSIONS

In this letter we introduced boundary control barrier functionals for PDEs, derived constraints for state invariance, and formulated a convex optimization-based control policy, providing sufficient conditions for optimal boundary controllers. While the regularity of the controller will be analyzed in future work, simulations seem to suggest that such property holds. Moreover, although in this study we focus on traffic flow dynamics with specific control objectives, the proposed approach lends itself to be extended to stability and invariance of generic PDEs.

REFERENCES

[1] R. Vazquez, J. Auriol, F. Bribiesca-Argomedo, and M. Krstic, "Backstepping for Partial Differential Equations," arXiv preprint arXiv:2410.15146, 2024.
[2] F. Troltzsch, ¨ Optimal control of partial differential equations: theory, methods, and applications. American Mathematical Soc., 2010, vol. 112.
[3] S. Daudin, "Optimal control of the Fokker-Planck equation under state constraints in the Wasserstein space," Journal de Mathematiques Pures ´ et Appliquees ´ , vol. 175, pp. 37–75, 2023.
[4] Y. S. Ledyaev and E. D. Sontag, "A Lyapunov characterization of robust stabilization," Nonlinear Analysis-Series A Theory and Methods-Series B Real World Applications, vol. 37, no. 7, pp. 813–840, 1999.
[5] A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, "Control barrier functions: Theory and applications," in 2019 18th European control conference (ECC). IEEE, 2019, pp. 3420–3431.
[6] H. Hu and C. Liu, "On the Boundary Feasibility for PDE Control with Neural Operators," Proceedings of Machine Learning Research vol vvv, vol. 1, p. 27, 2025.
[7] J. Wang and M. Krstic, "Safe adaptive control of hyperbolic PDE-ODE cascades," arXiv preprint arXiv:2309.05596, 2023.
[8] Z. Li, N. B. Kovachki, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, A. Anandkumar, et al., "Fourier Neural Operator for Parametric Partial Differential Equations," in International Conference on Learning Representations, 2021.
[9] M. J. Lighthill and G. B. Whitham, "On kinematic waves II. A theory of traffic flow on long crowded roads," Proceedings of the royal society of london. series a. mathematical and physical sciences, vol. 229, no. 1178, pp. 317–345, 1955.
[10] P. I. Richards, "Shock waves on the highway," Operations research, vol. 4, no. 1, pp. 42–51, 1956.
[11] C. Bardos, A.-Y. LeRoux, and J.-C. Ned´ elec, "First order quasilin- ´ ear equations with boundary conditions," Communications in partial differential equations, vol. 4, no. 9, pp. 1017–1034, 1979.
[12] S. Blandin, X. Litrico, M. L. Delle Monache, B. Piccoli, and A. Bayen, "Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws," IEEE Transactions on Automatic Control, vol. 62, no. 4, pp. 1620–1635, 2017.

chiriconvexoptimizationboundarycontrol2025.md 36 KB Histórico Raw